Limits and Derivatives Class 11 NCERT Solutions

NCERT Solutions Class 11 Mathematics Chapter 13 Limits And Derivatives Download In Pdf

In this pdf file you can see answers of following Questions

EXERCISE 13.1

Evaluate the following limits in Exercises 1 to 22.

Question 1. 3 lim 3 x x → +

Question 2. π lim 22 x 7 x →

Question 3. 2 1 limπ r r →

Question 4. 4 lim 4 3 x 2 x → x + −

Question 5. 10 5 1 lim 1 x 1 x x → − x + + −

Question 6. ( )5 0 1 1 lim x x → x + −

Question 7. 2 2 2 lim 3 10 x 4 x x → x − − −

Question 8. 4 3 2 lim 81 x 2 5 3 x → x x − − −

Question 9. 0 lim x 1 ax b → cx + +

Question 10. 1 3 1 1 6 lim 1 1 z z z → − −

Question 11. 2 1 2 lim , 0 x ax bx c a b c → cx bx a + + + + ≠ + +

Question 12. 2 1 1 lim 2 x 2 x →− x + +

Question 13. 0 lim sin x ax → bx

Question 14. 0 lim sin , , 0 x sin ax a b → bx ≠

Question 15. ( ) π ( ) sin π lim x π π x → x − −

Question 16. 0 lim cos x π x → − x

Question 17. 0 limcos 2 1 x cos 1 x → x − −

Question 18. 0 lim cos x sin ax x x → b x +

Question 19. 0 lim sec x x x →

Question 20. 0 lim sin , , 0 x sin ax bx a b a b → ax bx + + ≠ + ,

Question 21. 0 lim (cosec cot ) x x x → −

Question 22. π 2 lim tan 2π 2 x x → x −

Question 23. Find ( ) 0 lim x f x → and ( ) 1 lim x f x → , where ( ) ( ) 2 3, 0 3 1, 0 x x f x x x + ≤ = +>

Question 24. Find ( ) 1 lim x f x → , where ( ) 2 2 1, 1 1, 1 x x f x x x − ≤ = − − >

Question 25. Evaluate ( ) 0 lim x f x → , where ( ) | |, 0 0, 0 x x f x x x ≠ = =

Question 26. Find ( ) 0 lim x f x → , where ( ) , 0 | | 0, 0 x x f x x x ≠ = =

Question 27. Find ( ) 5 lim x f x → , where f (x) = | x | −5

Question 28. Suppose ( ) , 1 4, 1 , 1 a bx x fx x b ax x + <= = − > and if 1 lim x→ f (x) = f (1) what are possible values of a and b?

Question 29. Let a1, a2, . . ., an be fixed real numbers and define a function f (x) = (x − a1 ) (x − a2 )...(x − an ) . What is 1 lim x→a (x) ? For some a ≠ a1, a2, ..., an, compute lim x→a f (x).

Question 30. If ( ) 1, 0 0, 0 1, 0 x x f x x x x + < = = − > . For what value (s) of a does lim x→a f (x) exists?

Question 31. If the function f(x) satisfies ( ) 1 2 2 lim π x 1 f x → x − = − , evaluate ( ) 1 lim x f x → .

Question 32. If ( ) 2 3 , 0 , 0 1 , 1 mx n x f x nx m x nx m x + < = + ≤ ≤ + > . For what integers m and n does both ( ) 0 lim x f x → and ( ) 1 lim x f x → exist?

EXERCISE 13.2

Question 1. Find the derivative of x2 – 2 at x = 10.

Question 2. Find the derivative of 99x at x = l00.

Question 3. Find the derivative of x at x = 1.

Question 4. Find the derivative of the following functions from first principle.
(i) x3 − 27
(ii) (x −1)(x − 2)
(iii) 2 1 x
(iv) 1 1 x x + −

Question 5. For the function ( ) 100 99 2 .1 100 99 2 f x = x + x + + x + x + . Prove that f ′(1) =100 f ′(0) .

Question 6. Find the derivative of xn + axn−1 + a2 xn−2 + . . .+ an−1x + an for some fixed real number a.

Question 7. For some constants a and b, find the derivative of
(i) (x − a) (x − b)
(ii) ( )ax2 b 2 +
(iii) x a x b − −

Question 8. Find the derivative of xn an x a − − for some constant a.

Question 9. Find the derivative of
(i) 2 3 4 x −
(ii) (5x3 + 3x −1) (x −1)
(iii) x−3 (5 + 3x)
(iv) x5 (3 − 6x−9 )
(v) x−4 (3 − 4x−5 )
(vi) 2 2 1 3 1 x x x − + −

Question 10. Find the derivative of cos x from first principle.

Question 11. Find the derivative of the following functions:
(i) sin x cos x
(ii) sec x
(iii) 5sec x + 4cos x
(iv) cosec x
(v) 3cot x + 5cosec x
(vi) 5sin x − 6cos x + 7
(vii) 2tan x − 7sec x

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